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## Homework Statement

Show that if an elastic collision between a mass and a stationary target of equal mass is not head-on that the projectile and target final velocities are perpendicular. (Hint: Square the conservation of momentum equation, using ##p^2=p\cdot p##, and compare the resulting equation with the energy conservation equation.)

## Homework Equations

##p=mv##

##K=\frac{1}{2}mv^2##

## The Attempt at a Solution

So I followed the hint and got $$\overrightarrow{p_1^2}=\overrightarrow{p_1^{'2}}+\overrightarrow{p_2^{'2}}+2\left(\overrightarrow{p_1}\cdot \overrightarrow{p_2}\right)$$ Plugging in values I got $$m^2\overrightarrow{v_1^2}=m^2\overrightarrow{v_1^{'2}}+m^2\overrightarrow{v_2^{'2}}+2\left(m\overrightarrow{v}_1\cdot m\overrightarrow{v}_2\right)$$ $$\overrightarrow{v_1^2}=\overrightarrow{v_1^{'2}}+\overrightarrow{v_2^{'2}}$$ For the x- and y-components I got the following assuming the target traveled along the x-axis after the collision $$v_1^2cos^2(\theta)=v_1^{'2}cos^2(\theta ^{'})+v_2^{'2}$$ $$v_1^2sin^2(\theta)=v_1^{'2}sin^2(\theta ^{'})$$ This is where I got stuck and wasn't sure how to solve for ##v_1^{'}## or ##v_2^{'}##. I'm not entirely sure when or how to apply the equation for kinetic energy. Am I in the right direction or did I do something wrong?

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